View source for Runge theorem

A theorem on the possibility of polynomially approximating holomorphic functions, first proved by C. Runge (1885) (cf. also [[Approximation of functions of a complex variable|Approximation of functions of a complex variable]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r0828301.png" /> be a simply-connected domain in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r0828302.png" />-plane. Then any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r0828303.png" /> holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r0828304.png" /> can be approximated uniformly on compact sets inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r0828305.png" /> by polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r0828306.png" />. More precisely, for any compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r0828307.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r0828308.png" /> there is a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r0828309.png" /> with complex coefficients such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r08283010.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r08283011.png" />.

In other words: Any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r08283012.png" /> holomorphic in a simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r08283013.png" /> can be represented as a series of polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r08283014.png" /> converging absolutely and uniformly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r08283015.png" /> on compact sets inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r08283016.png" />.

An equivalent statement of Runge's theorem: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r08283017.png" /> be a compact set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r08283018.png" /> with connected complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r08283019.png" />; then any function holomorphic in a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r08283020.png" /> can be approximated uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r08283021.png" /> by polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r08283022.png" />. In this formulation, Runge's theorem is a special case of Mergelyan's theorem (cf. [[Mergelyan theorem|Mergelyan theorem]]).

The following theorem on rational approximation is also called Runge's theorem: Any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r08283023.png" /> holomorphic in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r08283024.png" /> can be uniformly approximated on compact sets inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r08283025.png" /> by rational functions with poles outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082830/r08283026.png" />.

Runge's theorem has many applications in the theory of functions of a complex variable and in functional analysis. A theorem analogous to Runge's theorem is valid for non-compact Riemann surfaces. An extension of Runge's theorem to functions of several complex variables is the Oka–Weil theorem (see [[Oka theorems|Oka theorems]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Markushevich,   "A short course on the theory of analytic functions" , Moscow  (1978)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  B.V. Shabat,   "Introduction of complex analysis" , '''1''' , Moscow  (1976)  (In Russian)</TD></TR></table>



====Comments====
For more about Runge's theorem and its generalizations, such as Walsh' theorem, the [[Keldysh theorem|Keldysh theorem]], the [[Lavrent'ev theorem|Lavrent'ev theorem]], see [[#References|[a6]]], [[#References|[a1]]], Chapt. VIII, and [[#References|[a3]]], Chapt. III, for the case of the complex plane; [[#References|[a2]]], Sect. 25, for the case of Riemann surfaces; and [[#References|[a4]]], Sect. 7, for the case of several complex variables.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.B. Burchel,   "An introduction to classical complex analysis" , '''1''' , Acad. Press  (1979)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  O. Forster,   "Lectures on Riemann surfaces" , Springer  (1981)  (Translated from German)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Gaier,   "Lectures on complex approximation" , Birkhäuser  (1987)  (Translated from German)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Wermer,   "Banach algebras and several complex variables" , Springer  (1976)  pp. Chapt. 13</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  W. Rudin,   "Real and complex analysis" , McGraw-Hill  (1978)  pp. 24</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A.I. Markushevich,   "Theory of functions of a complex variable" , '''3''' , Chelsea  (1977)  pp. Sect. 3.12  (Translated from Russian)</TD></TR></table>